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atoms in strong magnetic fields, maximum number of electrons bound, 
one-dimensional models
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\documentclass[12pt]{article}

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\title{A One-Dimensional Model for Many-Electron Atoms
 in Extremely Strong Magnetic Fields:  Maximum Negative Ionization}

\author{Raymond Brummelhuis \\
Department of Mathematics \\  Universit\'e de Reims, 
\\ F-51687 Reims, France \\ 
{\normalsize raymond.brummelhuis@univ-reims.fr}
\and Mary Beth Ruskai \thanks{supported by National Science
Foundation Grant DMS-94-08903} \\ Department of
Mathematics \\ University of Massachusetts  Lowell \\ Lowell,
MA  01854 USA \\ {\normalsize bruskai@cs.uml.edu} }

\date{18 December 1998}


\begin{document}

\maketitle

\begin{abstract}

We consider a one-dimensional model for many-electron atoms in
strong magnetic fields in which the Coulomb potential and
interactions are replaced by one-dimensional regularizations
associated with the lowest Landau level.  For this model we
show that the maximum number of electrons $N_{\max}$ satisfies a
bound of the form $N_{\max} < 2Z+1 + c \sqrt{B}$ where $Z$ denotes
the charge of the nucleus, $B$ the field strength and $c$ is a
constant. We follow Lieb's strategy in which convexity plays a
critical role.  For the case $N=2$ with fractional
nuclear charge, we also discuss the critical
value $Z_c$ at which the nuclear charge becomes too weak to
bind two electrons.

\end{abstract}

\pagebreak

\section {Introduction}

It is well-known that systems in strong magnetic fields behave 
like systems in one-dimension, i.e., a strong magnetic field  
confines the particles to Landau orbits orthogonal to the
field, leaving only their behavior in the direction of the field
subject to significant influence by a static potential.
Therefore, a better understanding of one-dimensional systems
is essential to understanding the behavior of systems in strong
magnetic fields.  Although many one-dimensional systems, including that of
 a hydrogen atom with a single electron \cite{AHS2,HR}, have been thoroughly
studied, relatively little is known about multi-particle atoms
confined to one dimension. 

In this paper we study the question of bounds on the maximum excess
negative charge using one-dimensional models of many-electron atoms.
Because our goal is an understanding of the behavior in
one-dimension, we do not deal with  the question of
accuracy of our one-dimensional models as approximations to, or
reductions from, real 3-dimensional atoms.  However, we sketch
such a reduction as motivation for the models considered.   

There is some question as to the proper replacement
for the Coulomb potential in one dimension \cite{HR}.  The potential
$V(x) = 1/|x|$ is so singular that the one-dimensional Hamiltonian
$ - \Delta - 1/|x|$ is not even essentially self-adjoint.  Fortunately,
an electron in a Landau orbit is attracted to a nucleus with
charge $Z$, not by a potential of the form $-Z/|x|$, but by a regularized
potential which is finite at the nucleus.  However, the
corresponding regularization of the electron-electron interaction
is more complicated unless both electrons have zero angular
momentum in the direction of the field.  In that case, the
regularized interaction has a simple form which is the basis for
our model.

There is an extensive literature on atoms in magnetic fields.  
Interest in atoms in extremely strong fields, which began in the 1970's
after the discovery of pulsars, has recently been renewed in the
1990's, culminating in the comprehensive work of 
Lieb, Solovej and Yngvason  (LSY) \cite{LSY1,LSY2, LSY3, Y}.
For a discussion of early work on approximations for atoms in extremely
strong magnetic fields we refer the reader to 
the insightful paper of Rau, Mueller, and Spruch \cite {RMS}
and to the review by Ruderman\cite{Rud}.
References to later work are given in the
introduction to  \cite{LSY2} and a summary of the work
of  LSY \cite{LSY2, LSY3} is
given in \cite{LSY1, Y}.  Rigorous work on atoms in magnetic
fields began with the work of Avron, Herbst and Simon (AHS)
\cite{AHS1, AHS2, AHS3}.  LSY \cite{LSY1,LSY2, LSY3, Y}
 not only analyzed extensions of Thomas-Fermi
theory in 5 distinct regions, but showed that
these regions suffice to give the correct asymptotics for the exact
Hamiltonian.  In particular, they showed that the maximum number
of electrons $N_{\max}(Z,B)$ which can be bound to a nucleus of
charge $Z$ in a constant magnetic field of strength $B$ 
satisfies $\liminf N_{\max}(Z,B)/Z \geq 2$ as $Z \rightarrow \infty$
and $B/Z^3 \rightarrow \infty.$  This should be compared with
asymptotic
neutrality \cite{LSST, FS, SSS}, i.e., 
$\lim_{Z \rightarrow \infty} N_{\max}(Z,0)/Z = 1$, for
atoms without magnetic fields.  We hope that the analysis of the
simple model in this paper is a modest first step toward a better
understanding of the mechanism by which extremely strong magnetic fields
bind an ``extra'' $Z$ electrons, as well as the conjectured converse
$N_{\max}(Z,B) \leq 2Z.$  




The full 3-dimensional Pauli Hamiltonian for an N-electron atom with 
nuclear charge $Z$ in a constant magnetic field of strength $B$,
acting on ${\cal H}^N$, the n-fold tensor product of 
$L^2({\bf R}^3) \otimes {\bf C}^2$, is
\begin{eqnarray}
  {\bf H}(N,Z,B,\alpha) = \sum_{j=1}^N \left[ |{\bf P}_j + {\bf A}|^2 + 
    {\bf \sigma}_j \cdot {\bf B} - Z/|{\bf r}_j| \right] +
     \sum_{j<k} \alpha/|{\bf r}_j - {\bf r}_k |
\end{eqnarray}
where ${\bf A}$ is a vector potential such that 
  ${\bf \nabla}  \times  {\bf A} = {\bf B}$ and $\alpha$ is a coupling
constant introduced for convenience in discussing scaling.  
If the spin-coupling term is omitted, it often suffices to consider
the corresponding scalar Hamiltonian, which we denote $H(N,Z,B,\alpha)$,
as an operator acting only on $[L^2({\bf R}^3)]^N$ or
the ``space"  portion of ${\cal H}^N$. We will choose
our coordinate system so that the field  ${\bf B} = (B,0,0)$ is in the
 x-direction and the gauge so that $2{\bf A} = {\bf B} \times {\bf r}$.
The Hamiltonian (1) satisfies the scaling relation
   ${\bf H}(N,Z,B,1) = B {\bf H}(N,Z B^{-1/2},1,B^{-1/2})$, 
i.e., we can scale out 
the field strength by replacing the nuclear charge $Z$ by $Z B^{-1/2}$
and reducing the electron-electron interaction by $B^{-1/2}$.
Alternatively, we could have included the electron charge unit  $e$
explicitly and replaced $e$ by $e B^{-1/4}$.


We let $E_0(N,Z,B,\alpha)$ denote the infimum of the spectrum of
the scalar Hamiltonian $H(N,Z,B,\alpha)$ defined above.
It is well-known that the spectrum of the
``free'' Hamiltonian  $|{\bf P}_j + {\bf A}|^2$ is $[B,\infty]$,
that the spectrum of $|{\bf P}_j + {\bf A}|^2 + {\bf \sigma}_j \cdot {\bf B}
  = [{\bf \sigma}_j \cdot ({\bf P}_j + {\bf A})]^2 $ is $[0,\infty)$
and that the lowest Landau level has energy $B$ with infinite
degeneracy indexed by $m = 0, 1, 2, 3, .... $ corresponding to
angular momentum $-m$ quantized in the field direction.
Therefore, the continuous spectrum of 
  $H(N,Z,B,\alpha)$ is $[ B + E_0(N-1,Z,B,\alpha), \infty)$, i.e.,
the continuum begins at $B$ plus the ground state energy for $N-1$ electrons.
Thus the question of whether or not $H(N,Z,B,\alpha)$ has a bound state
is determined by whether or not some test function $\Psi$ satisfies
\begin{eqnarray}
  \langle \Psi, H(N,Z,B,\alpha) \Psi \rangle  <  
    [B + E_0(N-1,Z,B,\alpha)] \| \Psi \|^2.
\end{eqnarray}
For the full Hamiltonian  ${\bf H}(N,Z,B,\alpha)$, the continuous
spectrum is $[E_0(N-1,Z,B,~\alpha), \infty)$ which implies that 
${\bf H}(N,Z,B,\alpha)$ has a bound state if and only if there is a
${\bf \Psi} \in {\cal H}^N$ for which
\begin{eqnarray}
  \langle {\bf \Psi}, {\bf H}(N,Z,B,\alpha) {\bf \Psi} \rangle  
    <  E_0(N-1,Z,B,\alpha)  \| {\bf \Psi} \|^2.
\end{eqnarray}
For the remainder of this paper we will omit explicit consideration
of spin [although we will be able to draw some conclusions indirectly.
See the remark after equation (\ref{varequivspin}).]  
Our methods cannot handle the   ${\bf \sigma}\cdot {\bf B}$
term explicitly and the inclusion of spin in the wave function does not
affect the remaining results in any essential way.  


We now consider the  the 3-dimensional energy minimization
problem restricted to functions whose behavior orthogonal to the
field is described entirely by product functions in which all electrons
are confined to the lowest Landau level, i.e we restrict to
 N-electron functions $\Psi$ of the form 
       $\Psi_{m_1 \ldots m_N} = 
      \Phi(x_1 \ldots x_n) \prod_{k=1}^N  \gamma_{m_k}^B(r_k,\theta_k)$
where  
  $$\gamma_m^B(r,\theta) = 
        [\pi m!]^{-1/2}B^{(m+1)/2} r^m e^{-Br^2/2}e^{-i m \theta}$$
denotes the Landau level with energy $B$ and angular momentum $-m.$
(Note that we are using cylindrical coordinates $(x,r,\theta)$ with
$r = \sqrt{y^2 + z^2}$ so that $|{\bf r}| = \sqrt{x^2 + r^2}$.)
 Such expectations satisfy
\begin{eqnarray} 
\lefteqn{\langle \Psi_{m_1 \ldots m_N}, H(N,Z,B,1) \Psi_{m_1 \ldots m_N} 
    \rangle} ~~~ \nonumber \\
  & = &   \langle \Psi_{m_1 \ldots m_N},  
       \widehat{H}(N,Z,B,1) \Psi_{m_1 \ldots m_N} \rangle  + NB   \\
   & = &  B \langle \Psi_{m_1 \ldots m_N},  
         \widehat{H}(N,ZB^{-1/2},1, B^{-1/2}) \Psi_{m_1 \ldots m_N} \rangle 
                       + NB  \nonumber
\end{eqnarray}
where 
\begin{eqnarray}
\widehat{H}(N,Z,B,\alpha) = \sum_{j=1}^N 
   \left[ - |P_j^x|^2  - Z/|{\bf r}_j| \right] +
     \sum_{j<k} \alpha/|{\bf r}_j - {\bf r}_k |.
\end{eqnarray}
For each fixed choice of $m_1 \ldots m_N$ the minimization problem
can be reduced to a one-dimensional problem for the Hamiltonian
$\widehat{H}_x^{m_1 \ldots m_N}(N,Z,B,\alpha)$ in which the kinetic
energy has the usual $-\frac{d^2}{dx^2}$ form and the
potentials $1/|{\bf r}_j|$ and $1/|{\bf r}_j - {\bf r}_k|$ 
are replaced by regularized potentials denoted by $V_{m_j}^B(x_j)$
 and    $W_{m_j,m_k}^B(|x_j - x_k|)$ respectively.
\begin{eqnarray}
  V_m^B(x) & = & \int\!\int |\gamma_{m}|^2 / |{\bf r}|dy dz \nonumber \\
     & = & \frac {B^{m+1}}{m!} \int_0^\infty 
        \frac{r^{2m} e^{-Br^2} }{\sqrt{x^2 + r^2}} r dr  \nonumber \\
     & = &  [ m!]^{-1} \int_0^\infty 
          \frac{u^m e^{-u} }{\sqrt{x^2 + u/B}} du  \label{vmdef} \\
    & = & \frac {2B^{m+1}}{m!} e^{Bx^2}\int_{|x|}^\infty  
             (t^2 - x^2)^m e^{-Bt^2} dt  \nonumber
\end{eqnarray}
which satisfies the scaling relation  
$V_m^B(x) = \sqrt{B}~ V_m^1(\sqrt{B}x)$ as one would expect from
the scaling properties of $H(N,Z,B,\alpha)$.
$W_{m_j,m_k}^B$ is defined analogously; we postpone discussion
of its explicit form.  That the regularizations $V_{m,B}(x)$
 are important for atoms in magnetic fields goes back at
least to Schiff and Snyder \cite{SS} in (1939) and played an
important role in the AHS study \cite {AHS2} of hydrogen. 
In the case $m = 0$ and $B = 1$, (6) can be rewritten as
\begin{eqnarray}
        V_0(x)  & = & \sqrt{\pi} e^{x^2} [1 - \hbox{erf}(x)]  \\
          & = &  \int_0^\infty  \frac{ e^{-u} }{\sqrt{x^2 + u}} du
                  = 2 e^{x^2}\int_x^\infty  e^{-t^2} dt \nonumber
\end{eqnarray}
A comparison with the usual Coulomb potential is given in Figure \ref{fig:V0}.
      
\begin{figure}
\vskip4cm
\hskip4cm\psfig{figure=magfinfg1.ps,height=1.2in,width=1.2in}
\vskip-0.5cm
\caption{Comparison of the potentials (reading from the top down)
$-\frac{1}{|x|+1}$, $-V_0(x)$, and $-\frac{1}{|x|}$}
\label{fig:V0}
\end{figure}
    
We now describe the correspondence between a restricted
three-dimensional minimization problem for a Hamiltonian with
 a magnetic field, namely,
\begin{eqnarray}
E_0^{m_1 \ldots m_N}(N,Z,B,\alpha) =  \inf_{\Psi_{m_1 \ldots m_N}} 
    \langle \Psi_{m_1 \ldots m_N}, H(N,Z,B,\alpha) \Psi_{m_1 \ldots m_N} \rangle
\end{eqnarray}
and the one-dimensional minimization problem 
\begin{equation}
  E_{0,x}^{m_1 \ldots m_N}(N,Z,B,\alpha) =  \inf_{\Phi} 
     \langle \Phi, \widehat{H}_x^{m_1 \ldots m_N}(N,Z,B,\alpha) \Phi \rangle
\end{equation}
where
\begin{equation}
  \widehat{H}_x^{m_1 \ldots m_N}(N,Z,B,\alpha) =
    \sum_{j=1}^N \left[ -\frac{d^2}{dx_j^2} -Z V_{m_j}^B(x_j) \right] +
        \alpha \sum_{j< k} W_{m_j,m_k}^B(|x_j - x_k|)
\end{equation}
in which no magnetic field is explicitly present.   Since
\begin{eqnarray} 
  \langle \Psi_{m_1 \ldots m_N}, H(N,Z,B,\alpha) \Psi_{m_1 \ldots m_N} \rangle
         =  \langle \Phi,  
       \widehat{H}_x^{m_1 \ldots m_N}(N,Z,B,\alpha) \Phi \rangle  + NB    
\end{eqnarray}
we can conclude that
\begin{eqnarray} 
 \langle \Psi_{m_1 \ldots m_N}, H(N,Z,B,\alpha) \Psi_{m_1 \ldots m_N} \rangle
   >  E_0^{m_1 \ldots m_{N-1}}(N-1,Z,B,\alpha)   + B   \nonumber \\ 
    \iff  
  \langle \Phi, \widehat{H}_x^{m_1 \ldots m_N}(N,Z,B,\alpha) \Phi \rangle
      >  E_{0,x}^{m_1 \ldots m_{N-1}}(N-1,Z,B,\alpha).
\end{eqnarray}
Thus, we have reduced the problem of determining whether or not the 
somewhat artificial problem of whether or not an atom in magnetic
field  $B$ whose electrons have prescribed angular momentum corresponding
to $m_1 \ldots m_N$ has a bound state to that of whether or not a
related one-dimensional system has a bound state. 
Although, we do not consider spin-coupling explicitly, it does follow
from (3) that we can also conclude that
\begin{eqnarray}\label{varequivspin} 
 \langle {\bf \Psi}_{m_1 \ldots m_N}, {\bf H}(N,Z,B,\alpha) 
         {\bf \Psi}_{m_1 \ldots m_N} \rangle
   >  E_0^{m_1 \ldots m_{N-1}}(N-1,Z,B,\alpha)   \nonumber \\ 
    \iff  
  \langle \Phi, \widehat{H}_x^{m_1 \ldots m_N}(N,Z,B,\alpha) \Phi \rangle
      >  E_{0,x}^{m_1 \ldots m_{N-1}}(N-1,Z,B,\alpha).
\end{eqnarray}
where ${\bf \Psi}_{m_1 \ldots m_N}$ is chosen so that all components in
${\bf C}^2$ correspond to spin ``down". 
(This observation, which is a fortuitous consequence of the physical fact
that the coefficient of ${\bf \sigma} \cdot {\bf B}$ is exactly $1$, was
pointed out to us by J.P. Solovej.)

We now consider the regularization of the interaction in the special
case in which $m_j = m_k = 0$.  Recall that 
$\gamma_0^B(y,z) = B \pi^{-1} e^{-B(y^2 + z^2)/2} = B \pi^{-1} e^{-Br^2/2}.$
\begin{eqnarray}
  W_{0,0}^B(|x_1 - x_2|) & = & \int\!\int  dy_1 dz_1\int\!\int  dy_2 dz_2 
    \frac{|\gamma_0^B(y_1,z_1)|^2|\gamma_0^B(y_2,z_2)|^2}
         {\sqrt{(x_1 - x_2)^2 + |(y_1,z_1) - (y_2,z_2)|^2} } \nonumber \\
   & = &   B \int_o^{\infty} dt  
        \frac{2t e^{-Bt^2}}{\sqrt{(x_1 - x_2)^2 + 2t^2} }  \\
   & = & \frac{1}{\sqrt{2}}  V_0^B\left(\frac{|x_1 - x_2|}{\sqrt{2}}\right)
       \nonumber
\end{eqnarray}
where we have made the change of variables 
\begin{eqnarray*}
{\bf s} = \frac{1}{\sqrt{2}} [(y_1,z_1) + (y_2,z_2)] ,~~~ 
      {\bf t} = \frac{1}{\sqrt{2}} [(y_1,z_1) - (y_2,z_2)]
\end{eqnarray*} 
with $s = |{\bf s}|,~ t = |{\bf t}|$, and used the fact that
\begin{eqnarray*}
           |\gamma_0^B(y_1,z_1)|^2 |\gamma_0^B(y_2,z_2)|^2 
& = & B^2\pi^{-2} e^{-B(r_1^2 + r_2^2)} \\
  & = &   B^2\pi^{-2}e^{-B(s^2 + t^2)}   
= |\gamma_0^B({\bf s})|^2  |\gamma_0^B({\bf t})|^2.
\end{eqnarray*}
Thus, the exceedingly simple relation  
$W_{0,0}^B(|x_1 - x_2|) = 2^{-1/2}V_0^B(2^{-1/2}|x_1~-~x_2|)$
follows from the invariance of 
    $|\gamma_0^B({\bf s})|^2  |\gamma_0^B({\bf t})|^2$
under the transformation of
   ${\bf s}, {\bf t}$ to $({\bf s} \pm {\bf t})/\sqrt{2}$.
This unusual invariance, corresponding to the mixing of coordinates
of two particles, will not hold if $m \neq 0$.
Symmetrizing the product or replacing 
$\Psi_{m_1 \ldots m_N}$ by an arbitrary
element of the projection onto the lowest Landau level, 
would require consideration of exchange terms as well.
Therefore, we will only study models corresponding to constraining
all electrons to have angular momentum zero.  Since $\Psi_{0 \ldots  0}$
is then symmetric with respect to exchange of $(y_j,z_j)$ with   
$(y_k,z_k)$, it will have the same permutational symmetry as
$\Phi(x_1 \ldots x_N)$.  



Despite the severity of the restriction to $m = 0$, our model seems
well-suited to study of the bounds on the negative ionization.
Integration by parts of (\ref{vmdef}) easily yields 
\begin{eqnarray}
  V_{m+1}^B(x) \leq  V_m^B(x) \leq  V_0^B(x) \leq   1/|x|
\end{eqnarray}
with the difference greatest at the origin and all potentials
satisfying  $V_m^B(x) \approx  1/|x|$ for large $x$.  Moreover,
for any choice of $m_j, ~ m_k$ whenever $|x_j - x_j|$ is large,
 $W_{m_j,m_k}^B(|x_j - x_j|) \approx  1/|x_j - x_j|$ as well.
Thus it
appears unlikely that placing some electrons in Landau levels
with $m \neq 0$ will allow binding if $m = 0$ does not.

With this heuristic background, we study models for N-electron atoms
in one-dimension corresponding to Hamiltonians of the form
\begin{eqnarray}\label{eq:Ham1dim}
  h(N,Z,M) & = &  M \widehat{H}_x^{0 \ldots 0}(N,Z,M^{-2},1) \\
    & = & \sum_{j=1}^N \left[ - \frac{1}{M} \frac{d^2}{dx^2} -Z V_0(x_j) \right] +
         \sum_{j< k} \frac{1}{\sqrt{2}} 
              V_0\left(\frac{|x_j - x_k|}{\sqrt{2}}\right) \nonumber   
\end{eqnarray}
with the ``mass'' $M$ proportional to $B^{-1/2}$.  Because of the
scaling relation $H(N,Z,B,1) = B H(N,ZB^{-1/2},1,B^{-1/2})$,
the only role of the field strength in the one-dimensional
situation, is to reduce the mass by a factor of $B^{-1/2}$.  Thus,
for simplicity, we have set both $B = 1$ and the coupling constant
$\alpha = 1$ leaving the field strength implicit in the mass $M$.

 As observed in \cite{AHS2}, the regularized potentials satisfy
\begin{eqnarray}
  \frac{1}{\sqrt{(m+1)B} + |x|} \leq \frac{1}{\sqrt{(m+1)B + |x|^2}}
      \leq  V_m^B(x) \leq  \frac{1}{|x|}.
\end{eqnarray}
The cut-off potential $V_{\hbox{\scriptsize{cut}}}(x) = 1/(|x| + 1)$ is also of some
interest.  Haines and Roberts \cite{HR} have given explicit solutions to the
eigenvalue problem for the hydrogenic Hamiltonian
$ - \Delta - V_{\hbox{\scriptsize{cut}}}(x)$.  By the above remark,
$V_{\hbox{\scriptsize{cut}}}(x) \leq V_0^1(x) \leq  1/|x|$.

In sections 2 and 3, we study
Hamiltonians of the form (\ref{eq:Ham1dim}) 
in the two cases $V = V_{\hbox{\scriptsize{cut}}}$ 
and $V = V_0$.  For both models we show that the existence of a
bound state implies  $N < 2Z+1+c \sqrt{B}$.

\section {Two-electron systems}

We now discuss in more detail the behavior of the one-dimensional
Hamiltonian (\ref{eq:Ham1dim}) when the number of electrons is
$N = 2$.  Although our discussion is descriptive and non-rigorous,
we believe the insights are useful.  The two-electron Hamiltonian
can be written in the form.
\begin{eqnarray}\label{eq:Ham2elec}
  h(2,Z,B^{-1/2}) = - \sqrt{B} \left[ \frac{d^2}{dx_1^2} +
  \frac{d^2}{dx_2^2} \right] + W(x_1,x_2)
\end{eqnarray}
where
\begin{eqnarray} \label{eq:Wxypot}
 W(x_1,x_2) = -Z V_0(x_1)  -Z V_0(x_2)
      + \frac{1}{\sqrt{2}}  V_0\left(\frac{|x_1 - x_2|}{\sqrt{2}}\right)
\end{eqnarray}

Consider a classical system of three particles on a line with charges
$-1$, $+Z$, $-1$ interacting with the usual inverse square Coulomb 
force (or, equivalently, $q_i q_j/|x_i- x_j|$ potential).
The only potentially stable configuration is one in which
the particles are arranged symmetrically with
the two negatively charged
particles on opposite sides of, and the same distance from, the positive
"nucleus" as shown in Fig. 1.  Even in this case,
the system is stable only for $Z = 1/4$.  For
$Z < 1/4$, both ``electrons" will move off to infinity, while for
$Z > 1/4$, the ``electrons'' collapse into the center.  This suggests
that for $Z > 1/4$ one might be able to show that the system binds
by using a trial function $\Psi_a$ in which the two electrons are localized 
on opposite sides of the nucleus at a distance $a$ sufficiently far
from the center that $V_0(\pm a) \approx 1/a$.  (Permutational
symmetry is not relevant; if such a trial function binds, then a
bosonic system will also bind.)   This picture would require
that $\Delta x_i < a$ for each electron so that, by the uncertainty
principle, the kinetic energy satisfies 
$\langle \Psi_a [- \frac{d^2}{dx_1^2}] \Psi_a \rangle > 1/a^2$.
Thus we estimate
\begin{eqnarray}
   \langle \Psi_a h(2,Z,B^{-1/2}) \Psi_a \rangle  >
      2 \frac {\sqrt{B}}{a^2}  - \frac{2Z}{a} + \frac{1}{2a}.
\end{eqnarray}
Minimizing over $a$ yields $a_{\min} = 2 \sqrt{B}/(Z - \frac{1}{4})$
and
\begin{eqnarray}
   \langle \Psi_a h(2,Z,B^{-1/2}) \Psi_a \rangle >
      - \frac {(Z - \frac{1}{4})^2}{2 \sqrt{B}}.
\end{eqnarray}
The continuous spectrum begins at the ground state energy of
 the corresponding one-electron Hamiltonian 
$h(1,Z,B^{-1/2}) = - \sqrt{B} \frac{d^2}{dx^2} -Z V_0(x)$
which Avron, Herbst and Simon \cite{AHS2} showed is given asymptotically
(for large $B$) by   $E_0(1,Z,B^{-1/2}) =
 - \frac{Z^2}{\sqrt{B}} \left( \log \frac{Z^2}{\sqrt{B}} \right)^2$
Thus, binding would require
\begin{eqnarray}
 (Z - \frac{1}{4})^2 > 2 Z^2  \left( \log \frac{Z}{\sqrt{B}} \right)^2.
\end{eqnarray}
This is obviously false for large $B$, which suggests that the
uncertainty principle prevents the corresponding one-dimensional
system from binding, even when the classical system collapses.
Because $M \sim B^{-1/2}$, large field strength corresponds to small
mass, i.e., as the field strength increases, the electrons become 
``lighter" and more difficult to localize.  Thus we must seek a
different mechanism to explain binding of a one-dimensional
two-electron atom.

Fortunately, our model provides an alternative mechanism for binding.
The  regularization of the potential
at the origin combined with the effective reduction of the interaction
by $1/\sqrt{2}$, permits both electrons to be close to the 
nucleus for $Z > 1/2^{3/2}$ in the sense that 
$-2ZV_0(0) + 2^{-1/2}V_0 < 0$.  Of course, the uncertainty
principle also precludes binding with a trial function in which
both electrons are exactly at the center.  Nevertheless,
we believe that the mechanism for binding is that
the effective reduction in the interaction permits the
two electrons to overlap strongly near the nucleus.


\begin{figure}
\vspace*{7cm}
\hspace*{1.5cm}\psfig{figure=magfinfg2A.ps,height=1in}
\hspace*{2.5cm}\psfig{figure=magfinfg2B.ps,height=1in}
\vskip-3cm
\caption {$W(x,-x)$ and $W(x,0)$ for (reading from the top 
down)  $Z~=~0.25, 0.3, 0.35, 0.4, 0.5, 0.6, 0.7, 0.8$.   
The left  graph, $W(x,-x)$, describes the potential when the electrons 
are on opposite sides of the nucleus with each a distance  
$x$ away. The right graph, $W(x,0)$, describes the potential 
when one electron is fixed at the origin; the left asymptote 
then corresponds to the other electron moving to infinity.}
\label{fig:Wxylines} \end{figure}

The Hamiltonian (\ref{eq:Ham2elec}) can be regarded as describing
either two particles in one-dimension or one particle in the
field of the two-dimensional potential $W(x,y)$ given by 
(\ref{eq:Wxypot}).  Regardless of our viewpoint, the continuous
spectrum will begin at the AHS \cite{AHS2} estimate of 
$ - \frac{Z^2}{\sqrt{B}} \left( \log \frac{Z^2}{\sqrt{B}} \right)^2$.
Since this is close to zero for large $B$, we will regard
$W(x,y)$ as attractive where it is negative and repulsive
where it is positive.
One might expect $W(x,y)$ to have its minimum on the line $y = -x$,
corresponding to electrons on opposite sides of the nucleus.  The
actual situation, which is described below, is more complex.
Plots of $W(x,-x)$ and $W(x,0)$ are shown in Fig. 2 for 
$Z = 0.25, 0.3, 0.35 \approx \frac{1}{2\sqrt{2}}, 0.4, 0.5,
 0.6, 0.7 \approx  \frac{1}{\sqrt{2}}, 0.8.$   The graph of
$W(x,0)$ is deliberately asymmetric so that the left asymptote,
$W(-\infty,0)$ can be compared to the minimum of $W(x,-x)$
at $x = 0.$  We now discuss this behavior in more detail,
noting that all the qualitative
features can also be verified analytically.
\begin{itemize}
\item[$\bullet$] $\frac{1}{4} < Z < \frac{1}{2\sqrt{2}} : $   
The potential does have a pair of weak minima along the line $y = -x$;
however, closer examination of the full two-dimensional 
potential shows that these are not true minima, but saddle
points for $W(x,y)$.  The potential is repulsive at the origin
and only weakly attractive elsewhere.
\item[$\bullet$] $\frac{1}{2\sqrt{2}} < Z < \frac{1}{2} : $ The potential is now
attractive at the origin.  However, as above, the two weak
minima on the line $y = -x$ correspond to saddle points of $W(x,y)$.
\item [$\bullet$] $\frac{1}{2} < Z  < \frac{1}{\sqrt{2}} : $ The two weak   
minima on the line $y = -x$ have now coalesced into a true
(two-dimensional) minimum at the origin.  However, this is only a local minimum.
$W(0,0) > W(0,\infty)$ so that the energy will decrease if one of 
the particles remains at the origin while the other goes off to infinity.
Fig. 2 suggests that the minimum along the $x=0$ and $y=0$ lines
is too shallow to prevent one of the electrons from tunnelling
through to infinity.  Thus, we expect resonances, but not bound
states in this region.
\item[$\bullet$] $ Z > \frac{1}{\sqrt{2}} : $  The potential has a true
minimum at the origin.   
\end{itemize}   
   



\begin{figure}
%%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
\vskip3.5cm
%%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
\hskip2in\psfig{figure=magfinfg3.ps,height=1in}
\vskip-0.5cm
\caption{Saddle point of $W(x,y)$ for $Z=0.4$ in the quadrant $x < 0$, $y > 0.$
 \label{fig:saddle}}
\end{figure}

Thus, the behavior of the potential $W(x,y)$ strongly suggests
that, at least for sufficiently large $B$,
binding occurs for $Z > \frac{1}{\sqrt{2}} \approx 0.7$ and that 
the mechanism which permits this is the reduction in strength
of the repulsion by $1/\sqrt{2}$ which permits both electrons to
simultaneously sit near the nucleus.  The behavior of the
regularized potential model for a two-electron atom in one dimension
seems to be very different from that of classical Coulomb particles
confined to a line.  The binding cut-off of $Z_c \approx 0.7$ 
represents a very modest ``binding enhancement'' when compared
to the value of $Z_c \approx 0.9$ obtained in \cite{BFHM}
for a two-electron atom in three dimension in the absence of a
magnetic field. [An ``enhancement'' to the level of
$2 = N  > 2Z+1$ would require $Z_c < 0.5$.]
\bigskip 


\section {Kinetic Energy and Lieb's Strategy}

We now describe the elegant strategy used by Lieb \cite{Lb} 
(see also section 3.8 of \cite{CFKS}) to show that for
real atoms the maximum number of electrons that can be bound is
less than $2Z+1$ irrespective of permutational symmetry.  The essence of
Lieb's argument is to show that one can replace the variation over
N-electron wave functions $\Psi({\bf r}_1, \ldots {\bf r}_N)$ by
a variation over density matrices of the form  
\begin{eqnarray}
\lefteqn{\Gamma({\bf r}_1, \ldots {\bf r}_N ; {\bf s}_1, \ldots {\bf s}_N)} \\
~~ & = & \sum_{j=1}^N 
    \overline{[\nu({\bf r}_j)]^{1/2} \Psi({\bf r}_1, \ldots {\bf r}_N)} 
[\nu({\bf s}_j)]^{1/2} \Psi({\bf s}_1, \ldots {\bf s}_N) 
   \nonumber
\end{eqnarray}
where $\nu({\bf r}) > 0$ is a strictly positive function
which will be chosen later. 
Note that, even though the functions $\Psi \nu_j^{1/2}$ 
are not orthonormal,
  $\T \Gamma A = \sum_j 
     \langle [\nu({\bf r}_j)]^{1/2} \Psi, A [\nu({\bf r}_j)]^{1/2} \Psi \rangle$
for any operator $A$.
Despite the introduction of the function $\nu$,
which will be chosen later, the $\Psi$  used in
$\Gamma$ satisfies the same symmetry or domain
constraints as the original minimization problem.

It will be useful to describe Lieb's argument in the rather
general situation of an N-particle Hamiltonian, $H_N$ with
the structure $H_N = H_{N-1}^j  + K_j + V_{\textstyle{\hbox{\scriptsize{int}}}}^j$ where
$K_j = |{\bf P}_j + {\bf A}|^2$ denotes the kinetic energy,
$H_{N-1}^j$ is the $N-1$-electron Hamiltonian in which the j-th
electron is omitted and $V_{\textstyle{\hbox{\scriptsize{int}}}}^j$ is a potential which
describes the interaction of the j-th electron with the rest
of the system.  Lieb studied $H_N = H(N,Z,0,1)$ while
we will consider $H_N =  h(N,Z,M)$.  In our case,
\begin{eqnarray}
V_{\textstyle{\hbox{\scriptsize{int}}}}^j = 
       -ZV(x_j) + \sum_{k \neq j} 2^{-1/2} V(2^{-1/2}|x_j - x_k|)
\end{eqnarray}
 
Now assume that $H_N$ actually has a bound state so that
there is a $\Psi_0$ satisfying $H_N \Psi_0 = E_0 \Psi_0$.  This
implies that for {\em any} function $\nu$ and {\em any} choice of $j$, 
\begin{eqnarray} 
    \nu({\bf r}_j) H_N \Psi_0({\bf r}_1, \ldots {\bf r}_N) 
    =  E_0 \nu({\bf r}_j) \Psi_0({\bf r}_1, \ldots {\bf r}_N) 
\end{eqnarray}
as well.  Hence
\begin{eqnarray}\label{eq:gamma.en} 
     \langle \Psi_0,  \nu({\bf r}_j) H_N \Psi_0 \rangle = 
   E_0   \langle \Psi_0,   \nu({\bf r}_j) \Psi_0\ \rangle .
\end{eqnarray}
Then if  $\Gamma_0$ is the density matrix of the above form corresponding
to the ground state $\Psi_0$, and $\nu_j$ denotes $\nu({\bf r}_j)$
\begin{eqnarray}\label{eq:gammavarprin}
\lefteqn  {E_0(N) \T \Gamma_0   =   
           E_0(N) \sum_{j=1}^N \langle  \Psi \nu_j\Psi \rangle} ~~~ \nonumber \\
 & = & \T \Gamma_0 H_N + \sum_{j=1}^N \left[ \langle\Psi \nu_j K_j \Psi \rangle 
          - \langle  \nu_j^{1/2} \Psi K_j \nu_j^{1/2}\Psi \rangle \right] 
     \nonumber  \\
 & = & \sum_j \langle  \nu_j^{1/2} \Psi,   \left [ H_{N-1}^j 
       + V_{\textstyle{\hbox{\scriptsize{int}}}}^j \right]   \nu_j^{1/2} \Psi \rangle  +
        \sum_{j=1}^N  \langle\Psi \nu_j K_j \Psi \rangle  \label{eq:Lbvar} \\
  & \geq &  E_0(N-1) \T \Gamma_0 +   \sum_j \left[
    \langle \nu_j^{1/2}\Psi, V_{\textstyle{\hbox{\scriptsize{int}}}}^j ~\nu_j^{1/2}\Psi \rangle
   + \langle\Psi \nu_j K_j \Psi \rangle \right]    \nonumber
\end{eqnarray}
The inequality in (\ref{eq:Lbvar}) above follows from the variational
principle for $H_{N-1}^j$.  Although  $\langle\Psi \nu_j K_j \Psi \rangle $
need not be real in general, a careful analysis of the argument
above shows that when $\Psi$ is an eigenstate of $H_N$ it
is real because it can be written as the difference of two real
quantities.  In Lieb's original formulation, $\nu$ was chosen
so that this real quantity satisfied
$\langle\Psi \nu_j K_j \Psi \rangle > 0.$  Then if 
$ \sum_j \langle \nu_j^{1/2}\Psi V_{\textstyle{\hbox{\scriptsize{int}}}}^j \nu_j^{1/2}\Psi \rangle > 0$
it follows that  $ E_0(N) \T \Gamma_0 >  E_0(N-1) \T \Gamma_0$
which contradicts the requirement for binding of
$ E_0(N) < E_0(N-1) $.  This reduces the problem of showing
that $H(N,Z,0,\alpha)$ has no bound states in the absence of
magnetic fields to that of showing that  
$ \sum_j \langle \nu_j^{1/2}\Psi V_{\textstyle{\hbox{\scriptsize{int}}}}^j \nu_j^{1/2}\Psi \rangle > 0.$
However, in a constant magnetic field the continuum begins
at $B + E_0(N-1)$ so that a similar reduction would require
the stronger condition  $\langle\Psi \nu_j K_j \Psi \rangle > B$.  

In order to study  $\langle\Psi \nu_j K_j \Psi \rangle$ in more
detail in different situations, it is convenient to
write $\nu_j$ as $g^2$ and observe that formally
\begin{eqnarray}\label{eq:kegena} 
\lefteqn{\langle \Psi g^2, |i\nabla_j + {\bf A}|^2 \Psi \rangle  
   = \langle [i\nabla_j + {\bf A}](g\Psi) g , 
          [i\nabla_j + {\bf A}](g \Psi) g^{-1} \rangle} \nonumber \\
  & = & 
  \langle [i\nabla_j + {\bf A}]g\Psi, [i\nabla_j + {\bf A}] (g\Psi) \rangle
       - \int |g\Psi|^2 \left|\frac{ \nabla g}{g}\right|^2    \\
    & & \pm ~2i~\Re \langle \Psi \nabla g,   [i\nabla_j + {\bf A}] (g\Psi) .
  \rangle   \nonumber
\end{eqnarray}
Note that the localization error,
$- \int |g\Psi|^2 \left|\frac{ \nabla g}{g}\right|^2 =   
\int |g\Psi|^2 \nabla g \cdot \nabla (g^{-1})$,
shows a certain symmetry between $g$ and $g^{-1}.$
If we now let  $g = \nu^{1/2}$, (\ref{eq:kegena}) implies
\begin{eqnarray}\label{eq:kegenb}
  \Re \langle\Psi \nu_j K_j \Psi \rangle =   \langle \Psi \nu_j^{1/2},
 \left[ K_j - \left| \frac{\nabla_j \nu_j}{2\nu_j} \right|^2 \right]
           \Psi \nu_j^{1/2} \rangle .
\end{eqnarray}
For the typical choice $\nu = [V({\bf r})]^{-1}$, one has
$\nabla \nu({\bf r})/\nu({\bf r}) = - \nabla V({\bf r})/V({\bf r})$
which yields
\begin{eqnarray}\label{eq:kegenc}
  \Re \langle\Psi \nu_j K_j \Psi \rangle =   \langle \Psi \nu_j^{1/2},
   \left[ K_j - \left| \frac{\nabla_j V_j}{2V_j} \right|^2 \right]
           \Psi \nu_j^{1/2} \rangle 
\end{eqnarray}
in accordance with the symmetry between $g$ and $g^{-1}$ noted above.
Because, as discussed above, we will only be concerned with
applications for which $\langle\Psi \nu_j K_j \Psi \rangle$ is real 
we will henceforth omit the $\Re$.  We now  discuss
several cases in more detail using ${\bf d}$ to denote the dimension
of the space on which $K_j$ acts.
(The original proof of Lieb used an argument which originated with   
Benguria (see Lemma 7.20 of \cite{LbRMP}) in the spherically symmetric   
case to show directly that   
$\Re \langle \nu \Psi (-\Delta) \Psi \rangle > 0$.  The variant
given here is due to Hoffman-Ostenhof \cite{HO2}.  For other strategies see
\cite{Baum,CFKS,Ich})
  
\noindent $\bullet~ {\bf A = 0, d \geq 3: }$ In this case we are
interested in  $\nu = [V({\bf r})]^{-1} $ for potentials, (particularly the
usual Coulomb potential $V({\bf r}) = 1/|{\bf r}|$) which
satisfy $\left| \frac{\nabla V({\bf r})}{V({\bf r})} \right| 
     \leq \frac{1}{|{\bf r}|}.$ Then $K = -\Delta$ and
(\ref{eq:kegenc}) becomes
\begin{eqnarray}\label{eq:ke3dim}
  \langle \Psi \nu_j K_j \Psi \rangle &=&
        \langle \Psi \nu_j (-\Delta_j) \Psi \rangle \nonumber \\
  & \geq &  \langle \Psi \nu_j^{1/2},  
        \left[ -\Delta_j - \frac{1}{4 |{\bf r}|^2}
              \right] \Psi \nu_j^{1/2} \rangle \geq 0
\end{eqnarray}

\noindent $\bullet{\bf A \neq 0, d = 3: }$  In this case we can only
conclude from (\ref{eq:kegenb})that
\begin{eqnarray}
  \langle\Psi \nu_j K_j \Psi \rangle & \geq &  B \|\Psi \nu_j^{1/2}\|^2
   -  \langle \Psi \nu_j^{1/2},  \left[ \frac{\partial^2}{\partial x_j^2} - 
     \left| \frac{\nabla_j \nu_j}{2\nu_j} \right|^2 \right] 
         \Psi \nu_j^{1/2} \rangle \\ 
   & \geq &  B \|\Psi \nu_j^{1/2}\|^2
  - \langle \Psi \frac{|\nabla_j \nu_j|^2}{4\nu_j}  \Psi  \rangle. \nonumber
\end{eqnarray}
We could instead have proceeded as in (\ref{eq:ke3dim}) above to obtain, 
$ \langle\Psi \nu_j K_j \Psi \rangle > 0$, but we cannot
conclude that 
$ \langle\Psi \nu_j K_j \Psi \rangle  \geq   B \|\Psi \nu_j^{1/2}\|^2.$
The kinetic energy
 $P_x^2 = \frac{\partial^2}{\partial x_j^2}$ in the field direction is 
not able to control a 3-dimensional potential, such as $1/{\bf r}^2$
arising from $\left| \frac{\nabla_j \nu_j}{2\nu_j} \right|^2 $.
Our one-dimensional models circumvents this problem
because the choice of product state involving Landau functions
ensured that the $P_y^2 + P_z^2$ terms took care of the $B$, leaving
$P_x^2$ to deal with a one-dimensional potential$[V'(x)/2V(x)]^2$ 
as described below.
   
\noindent$\bullet~$ {\bf A = 0, d=1:} In one dimension, ${\bf A}$ plays   
no role.  Although we will still choose
$\nu = [V(|r|)]^{-1} $ with potentials which satisfy
$\left|\frac{V^\prime(x)}{x}\right|^2 < \frac{1}{|x|^2}$ it is not true in
one or two-dimensions that $-\Delta - |2x|^{-2} > 0$.  Instead
we will treat 
$- \left| \frac{V^{\prime}}{2V} \right|^2$ as a potential in  
\begin{eqnarray}
 \langle \Psi \nu_j (-\Delta_j) \Psi \rangle =
    \langle \Psi \nu_j^{1/2},  \left[ -\frac{d^2}{dx_j^2} -  
    \left| \frac{V^{\prime}(x_j)}{2V(x_j)} \right|^2 \right] 
              \Psi \nu_j^{1/2} \rangle .
\end{eqnarray}
Now in one-dimension the everywhere negative potential 
$- \left| \frac{V^{\prime}}{2V} \right|^2 $ will always give rise to
a bound state of 
$ -\frac{d^2}{dx^2} - \left| \frac{V^{\prime}}{2V} \right|^2$.  
Thus we can only conclude
that  $\langle \Psi \nu_j (-\Delta_j) \Psi \rangle \geq
   -\epsilon \| \Psi \nu_j^{1/2} \|^2 $ where $ -\epsilon$ is the lowest
eigenvalue of this operator.  However, the bound 
\begin{eqnarray}
  \langle \Psi \nu_j (-\Delta_j) \Psi \rangle \geq
     - \omega \| \Psi \|^2
\end{eqnarray}
where $\omega = \sup_x | \nu^\prime(x)|^2 /4\nu(x)$
will be more useful.   Notice that, unlike the 3-dimensional case
where the lower bound is zero, when applying this result to
$H_N = h(N,Z,M)$ we will need
to take into account the fact the entire kinetic energy term
is multiplied by $1/M$ (or $\sqrt{B}$).



\section {Bound on the Maximum Negative Ionization}


We now apply Lieb's strategy, which yields $N_{\max}(Z,0) < 2Z+1$
for atoms without magnetic fields, to our one-dimensional models
for systems in strong magnetic fields.  We obtain the following
result.
\begin{thm}  The maximum number of electrons $N_{\max}$ for which
a Hamiltonian $h(N,Z,B^{-1/2})$ of the form (\ref{eq:Ham1dim}) has
a bound state satisfies
 $N_{\max} < 2Z+1 + c \sqrt{B}$ for some constant $c$.
\end{thm}
In the interesting case $B = O(Z^3)$ (which is the boundary of the   
LSY hyperstrong limit region in \cite{LSY2}), this yields a bound of the   
form $N_{\max} < 2Z+ c Z^{3/2}$, rather than  a linear
one of the form  $N_{\max} < cZ$ or the expected optimal
$N_{\max} < 2Z + o(Z)$.

We apply the strategy of Section 3 with $H_N = h(N,Z,M)$ and $\nu = 1/V$.
In this case, the analysis of expectations of 
$V_{\textstyle{\hbox{\scriptsize{int}}}}^j$
is straightforward because 
$ \langle \nu_j^{1/2} \Psi, V   \nu_j^{1/2} \Psi \rangle  =
       \langle \Psi, \Psi \rangle$ .
\begin{eqnarray}
\lefteqn{ \sum_j \langle  \nu_j^{1/2}\Psi
        V_{\textstyle{\hbox{\scriptsize{int}}}}^j \nu_j^{1/2}\Psi \rangle}~~~
     \nonumber \\  
 & = &   -Z \sum_j \| \Psi \|^2  + \sum_j \sum_{k \neq j}
      \langle \Psi, 2^{-1/2} V(2^{-1/2} |x_j - x_k|) V(x_j)^{-1} \Psi \rangle \\
  & = & -NZ \| \Psi \|^2 + \sum_{j < k} \langle \Psi, 2^{-1/2}     
      V(2^{-1/2} |x_j - x_k|)[ V(x_j)^{-1} + V(x_k)^{-1}] \Psi \rangle. \nonumber
\end{eqnarray}
Thus, {\em if} the potential satisfies 
\begin{eqnarray}\label{triineqsubs}
   2^{-1/2}V(2^{-1/2} |x_j - x_k|)[ V(x_j)^{-1} + V(x_k)^{-1}] > 1,
\end{eqnarray}
then
\begin{eqnarray}
 \sum_j \langle  \nu_j^{1/2} \Psi 
        V_{\textstyle{\hbox{\scriptsize{int}}}}^j \nu_j^{1/2} \Psi \rangle \geq
  [- NZ + N(N-1)/2 ] \| \Psi \|^2.
\end{eqnarray}

For the cut-off potential $V(x) = 1/(|x|+1)$, (\ref{triineqsubs})
follows easily from the triangle inequality since
\begin{eqnarray*} 
2^{1/2}[V(2^{-1/2} |w-x|)]^{-1} &=& |w-x| + 2^{1/2}  \\
   & \leq & |w| + 2^{1/2}  + |x| + 2^{1/2}  = [ V(w)]^{-1} + [V(x)^{-1}]
\end{eqnarray*}
For the regularized potential $V_0(x)$ we will use instead
the  convexity of $\nu_0(x) = [V_0(x)]^{-1}$.  (Because the proof 
\cite{W,SzW}
of this essential fact, although elementary, is rather delicate and not
readily accessible, we provide a sketch in the Appendix.)
Thus we find,
\begin{eqnarray}
    [ V_0(w)]^{-1} + [V_0(x)^{-1}] & \geq &  2 [V_0(|w-x|/2)]^{-1} \nonumber \\
      & \geq &  2^{1/2} [V_0(2^{-1/2}|w-x|]^{-1}
\end{eqnarray}
where the first inequality used $V_0(x) = V_0(|x|)$ as well as the
convexity of  $\nu_0(x) = [V_0(x)]^{-1}$ and  the second inequality follows
from
\begin{eqnarray*}
  V_m(2^{-1/2}y) 
   &=&  [m!]^{-1} \int_0^\infty \frac{t^m e^{-t}}{\sqrt{y^2/2 + t}}  ~dt \\
  & = & 2^{1/2} [m!]^{-1}\int_0^\infty  \frac{t^m e^{-t}}{\sqrt{y^2 + 2t}}  ~dt \\
  & \leq & 2^{1/2} V_m(y)  
\end{eqnarray*}
with  $y = 2^{-1/2} |w-x|$ and $m=0$.


Thus we can conclude that, for both the cut-off potential
$V_{\hbox{\scriptsize{cut}}}(x) = 1/(|x| + 1)$ 
and the regularized potential $V_0(x)$,
the one dimensional Hamiltonian $h(N,Z,M)$ satisfies
\begin{eqnarray}\label{eq:lastcond}
\lefteqn{\T \Gamma_0 h(N,Z,M)}~~~ \\ & \geq & E_0(N,Z,M) \T \Gamma_0 + 
      [- NZ + N(N-1)/2 -N \omega/M ]\| \Psi \|^2. \nonumber
\end{eqnarray}
The second term in (\ref{eq:lastcond}) will be positive if
$(N-1)/2 > Z + \omega/M.$ 
For both potentials,  
$\omega = \sup_x | \nu^\prime(x)|^2 /4\nu(x)$
is given by $\lim_{x \rightarrow 0} |\nu^\prime(x)|^2 /4\nu(x).$ 
This yields, $\omega_{\hbox{\scriptsize {cut}}} = 0.25$  for the cut-off
potential and $\omega_0 =  \pi^{-3/2} < 0.18 $ 
for the regularized potential $V_0(x).$   The latter follows
from  $\omega = \nu(0)^3 =  \pi^{-3/2}$ and the fact that,
as shown in the Appendix,
$|\nu^\prime(x)|^2 /4\nu(x) = \nu(x)[\nu(x)-|x|]^2$ is decreasing
for $x > 0$. 
Thus, in both cases, the second term in (\ref{eq:lastcond}) will 
be positive if $N > 2Z+1+1/2M$.  Since binding implies
 $ \T \Gamma_0 h(N,Z,M) \leq  E_0(N,Z,M) \T \Gamma_0 $
and $M$ is proportional to $B^{-1/2}$,
we have shown that one can find a constant $c$ such that
our model one-dimensional system does not have bound states if  
$N \geq 2Z+1+c\sqrt{B}$ or $N_{\max} < 2Z+1+c\sqrt{B}$.




Although this bound is not optimal,
it should be remembered that we are analyzing a Hamiltonian
with the structure
\begin{eqnarray} 
h(N,Z,B^{-1/2}) = h(N-1,Z,B^{-1/2}) + B^{1/2}K_j + 
    V_{\textstyle{\hbox{\scriptsize{int}}}}(x_j)
\end{eqnarray}
and that the factor $\sqrt{B}$ multiplying the kinetic energy is
something of a two-edged sword.  On the one hand, it raises the
energy of the effective one-electron Hamiltonian
$B^{1/2}K_1 + \int V_{\textstyle{\hbox{\scriptsize{int}}}}
|\psi(x_1,x_2,\ldots,x_N)|^2 dx_2 \ldots dx_N $; 
on the other, it multiplies any error arising from the kinetic
energy -- whether using Lieb's strategy or the Ruskai-Sigal
localization approach \cite{CFKS,Rusk1,Rusk2,Sig} -- by $\sqrt{B}.$
Since such correction terms are typically negative, the result
of such treatments is to perversely magnify the negative error associated
with a positive term otherwise expected to raise the energy.
\bigskip

\noindent{\bf Acknowledgment}  It is a pleasure to thank Professor
Pierre Duclos for many helpful discussions, including a critical
observation about the form of the interaction, and for his
hospitality during visits to the the Universit\'e de Toulon et du Var
and the Centre Physique Th{\'e}orique at Luminy-Marseille. 
We are also grateful to Professor Phillip Solovej for several helpful
comments and to Professor Elisabeth Werner for communicating
the results in \cite{SzW} and the strategy used to prove inequality
(\ref{ineq:g3Vg4}) in the Appendix.  
The second author [MBR] appreciated the hospitality 
at the Universit\'e de Reims and at Georgia Tech where parts of this
work were done.

\appendix\section {Convexity and Properties of $1/V_0(x)$}
  Because $ \nu(x) = 1/V_0(x)$ is nearly linear (see Figure \ref{fig:convex}),
the proof of its convexity is somewhat delicate.  It will follow from
the upper bound in the following pair
of inequalities, which are of some interest in their own right.
\begin{eqnarray}\label{ineq:g3nug4}
 \frac{3x + \sqrt{x^2+4}}{4} < \nu(x) <  \frac{2x + \sqrt{x^2+3}}{3}.
\end{eqnarray}
We now restrict attention to $x > 0$ and define
\begin{eqnarray}\label{eq:gkdef}
 g_k(x) = \frac{k}{(k-1)x + \sqrt{x^2+k}}
\end{eqnarray}
so that (\ref{ineq:g3nug4}) is equivalent to
\begin{eqnarray}\label{ineq:g3Vg4}
  g_4(x) > V_0(x) > g_3(x).
\end{eqnarray}
which is an improvement on the classical inequalities of Komatsu
\cite{IM}.  The upper bound and convexity were established independently 
by Wirth \cite{W} and by Szarek and Werner \cite{SzW}; the proof of
lower bound which follows was communicated to the authors by
E. Werner.  


\begin{figure}
\vskip3cm
\hskip4cm\psfig{figure=magfinfg4.ps,height=1in,width=1in}
\vskip-0.5cm
\caption{Comparison of $1/V_0(x)$ and $x$ for $x \geq 0$.}
\label{fig:convex}
\end{figure}
 

  We now observe that $V_0(0) = \sqrt{\pi}$ and $V_0$ satisfies the 
  differential equation
\begin{eqnarray}\label{diffeqV}
  V_0'(x) = 2[x V_0(x) -1]
\end{eqnarray} 
and look for analogous behavior for $g_k.$  
Since  $g_k^{\prime}(x) = -[g_k(x)]^2~
     \frac{x + (k-1)\sqrt{x^2+k}}{k~\sqrt{x^2+k} } $ and
    $ x g_k(x) -1 = \frac{-1}{x + \sqrt{x^2+k}}~g_k(x)$,
one can verify that
\begin{eqnarray}\label{ineq:deriv}
   \lefteqn{ g_k'(x) > 2[x g_k(x)  - 1] } ~~~~~~~~ \nonumber \\
        & \iff &  \frac{k}{\sqrt{x^2+k}}~
    \frac{(k-1)\sqrt{x^2+k} + x}{(k-1)x + \sqrt{x^2+k}} < 
      \frac{2k}{x + \sqrt{x^2+k}} \nonumber \\
           & \iff & (k-2) x^2 + k (k-3) < (k-2) x \sqrt{x^2+k} \nonumber
        \\ & \iff & x^2 (k-2)(k-4) + k(k-3)^2 < 0.
 \end{eqnarray}
 For $k=3$, we use the first equivalence to conclude from   
$x^2 < x \sqrt{x^2+3} $ that $g_3'(x) > 2[xg_3(x)  - 1] $;   
for $k=4$, we use the second to conclude
$k(k-3)^2 > 0 \Rightarrow  g_4'(x) < 2[xg_4(x)  - 1].$

For $k=3$, let $h_3(x) = V_0(x) - g_3(x)$. It is easily to verify that
$h_3(0) = \sqrt{\pi} - \sqrt{3} > 0$ and that   
$\lim _{x \to \infty } |h_3(x)| = 0 $. The first   
equivalence in (\ref{ineq:deriv}) implies
that $x^2 < x \sqrt{x^2+3} \Rightarrow g_3'(x) > 2[xg_3(x)  - 1]$
so that $h'_3(x) < 2x [V_0 (x) - g_3(x)] = 2x h_3(x) ~\forall x.$
Now suppose that for some $x= a$, $h_3(a) < 0$. Then $h'_3(a) < 0$ and
it follows that $h_3(x)$ is decreasing for $x > a$, which 
contradicts $\lim_{x \rightarrow \infty} h_3(x) = 0.$  This
proves the lower bound in (\ref{ineq:g3Vg4}).  The upper bound
is proved similary \cite{SzW} except that one now shows that
$h_4(x) = g_4(x) - V_0(x) > 0.$
It is interesting to note that the upper bound in
(\ref{ineq:g3Vg4})  is optimal, but the lower bound can be
improved \cite{RuW} to $g_{\pi}(x) < V_0(x).$
The tightness of these bounds is evident in Figure \ref{fig:comparebd}.


\begin{figure}
\vskip3cm
\hskip5cm\psfig{figure=magfinfg5.ps,height=1in,width=1in}
\vskip-0.5cm
\caption{Graphs showing $g_4(x) > V_0(x) > g_3(x)$ for $x > 0$.}
\label{fig:comparebd}
\end{figure}


In order to use this to establish the convexity of $\nu(x)$, 
we note (\ref{diffeqV}) implies that   
 $\nu(x) = [V_0(x)]^{-1}$ satisfies  
\begin{eqnarray}\label{diffeq}   
 \nu'(x) = 2\nu(x)[\nu(x) - x].   
\end{eqnarray}   
Then   
\begin{eqnarray}\label{deriv2}   
 \nu''(x) & =  &4 \nu\nu' -2 \nu'x -2\nu    
     = 4\nu(\nu-x)(2\nu-x) -2\nu \nonumber \\   
  & = & 8\nu [ (\nu - \textstyle{\frac{3}{4}} x)^2 - \textstyle{\frac{1}{16}}
            (x^2+4)]  \\   
   & = & 8\nu \left[ (\nu(x) - 1/g_4(x) + \textstyle{\frac{1}{4}} \sqrt{x^2+4})^2 - 
       (\textstyle{\frac{1}{4}} \sqrt{x^2+4})^2  \right]. \nonumber 
\end{eqnarray}
Since (\ref{ineq:g3nug4}) implies
$\nu(x) > 1/g_4(x)$, we can conclude that $\nu''(x)  > 0.$

To show that
$|\nu^\prime(x)|^2 /4\nu(x) = \nu(x)[\nu(x)-|x|]^2$  is decreasing for 
$x > 0$, we note that (\ref{diffeq}) yields
\begin{eqnarray*}
 \frac{d}{dx} \left[ \nu (\nu-x)^2  \right] & = &
   2 \nu (\nu - x) [3 \nu^2 - 4 \nu x + x^2 -1] \\
     & = & 6 \nu (\nu - x) \left[ (\nu - \frac{2x}{3})^2 - \frac{x^2 + 3}{9} 
     \right]  < 0.
\end{eqnarray*}
and (\ref{ineq:g3nug4}) implies that the expression in square brackets
is negative.

It is useful to have bounds on the extent to which
$\nu(x)$ and $V_0(x)$ deviate from $|x|$ and the Coulomb potential,
respectively, when $x$ is large.  The upper bound in (\ref{ineq:g3nug4})
implies 
$\left|\nu(x) - |x| \right| < g_4(x) - x \leq 1/2|x|$ 
while the lower bound yields 
$\left|V_0(x) - 1/|x| \right| < 1/x - g_3(x) < 1/2x^3.$  
The latter can also be proved directly from the integral
representation (\ref{vmdef}).

   
    
    
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\end{thebibliography}

 \end{document}
 

 
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2420 4092 l
S
NP 2179 4108 m 2259 4201 l
2339 4149 l
S
NP 2099 4161 m 2179 4249 l
2259 4201 l
S
NP 2019 4207 m 2099 4291 l
2179 4249 l
S
NP 1939 4246 m 2019 4324 l
2099 4291 l
S
NP 1859 4275 m 1939 4346 l
2019 4324 l
S
NP 1778 4289 m 1858 4354 l
1939 4346 l
S
NP 1698 4286 m 1778 4341 l
1858 4354 l
S
NP 1618 4256 m 1698 4302 l
1778 4341 l
S
NP 1538 4191 m 1618 4226 l
1698 4302 l
S
NP 1458 4077 m 1538 4098 l
1618 4226 l
S
NP 1377 3890 m 1458 3896 l
1538 4098 l
S
NP 1297 3600 m 1377 3587 l
1458 3896 l
S
NP 3221 3436 m 3301 3548 l
S
NP 3382 3480 m 3302 3367 l
S
NP 3301 3548 m 3382 3480 l
S
NP 3141 3505 m 3221 3616 l
3301 3548 l
S
NP 3061 3573 m 3141 3683 l
3221 3616 l
S
NP 2981 3642 m 3061 3749 l
3141 3683 l
S
NP 2901 3709 m 2981 3815 l
3061 3749 l
S
NP 2821 3776 m 2901 3881 l
2981 3815 l
S
NP 2740 3842 m 2820 3945 l
2901 3881 l
S
NP 2660 3907 m 2740 4008 l
2820 3945 l
S
NP 2580 3971 m 2660 4069 l
2740 4008 l
S
NP 2500 4033 m 2580 4128 l
2660 4069 l
S
NP 2420 4092 m 2500 4185 l
2580 4128 l
S
NP 2339 4149 m 2419 4238 l
2500 4185 l
S
NP 2259 4201 m 2339 4287 l
2419 4238 l
S
NP 2179 4249 m 2259 4331 l
2339 4287 l
S
NP 2099 4291 m 2179 4368 l
2259 4331 l
S
NP 2019 4324 m 2099 4397 l
2179 4368 l
S
NP 1939 4346 m 2019 4413 l
2099 4397 l
S
NP 1858 4354 m 1938 4414 l
2019 4413 l
S
NP 1778 4341 m 1858 4395 l
1938 4414 l
S
NP 1698 4302 m 1778 4347 l
1858 4395 l
S
NP 1618 4226 m 1698 4261 l
1778 4347 l
S
NP 1538 4098 m 1618 4122 l
1698 4261 l
S
NP 1458 3896 m 1538 3907 l
1618 4122 l
S
NP 1377 3587 m 1457 3583 l
1538 3907 l
S
NP 3301 3548 m 3381 3652 l
S
NP 3462 3585 m 3382 3480 l
S
NP 3381 3652 m 3462 3585 l
S
NP 3221 3616 m 3301 3719 l
3381 3652 l
S
NP 3141 3683 m 3221 3785 l
3301 3719 l
S
NP 3061 3749 m 3141 3850 l
3221 3785 l
S
NP 2981 3815 m 3061 3914 l
3141 3850 l
S
NP 2901 3881 m 2981 3978 l
3061 3914 l
S
NP 2820 3945 m 2900 4040 l
2981 3978 l
S
NP 2740 4008 m 2820 4101 l
2900 4040 l
S
NP 2660 4069 m 2740 4161 l
2820 4101 l
S
NP 2580 4128 m 2660 4217 l
2740 4161 l
S
NP 2500 4185 m 2580 4272 l
2660 4217 l
S
NP 2419 4238 m 2499 4322 l
2580 4272 l
S
NP 2339 4287 m 2419 4368 l
2499 4322 l
S
NP 2259 4331 m 2339 4409 l
2419 4368 l
S
NP 2179 4368 m 2259 4442 l
2339 4409 l
S
NP 2099 4397 m 2179 4466 l
2259 4442 l
S
NP 2019 4413 m 2099 4477 l
2179 4466 l
S
NP 1938 4414 m 2018 4473 l
2099 4477 l
S
NP 1858 4395 m 1938 4447 l
2018 4473 l
S
NP 1778 4347 m 1858 4392 l
1938 4447 l
S
NP 1698 4261 m 1778 4298 l
1858 4392 l
S
NP 1618 4122 m 1698 4149 l
1778 4298 l
S
NP 1538 3907 m 1618 3922 l
1698 4149 l
S
NP 1457 3583 m 1537 3585 l
1618 3922 l
S
NP 3381 3652 m 3461 3750 l
S
NP 3542 3684 m 3462 3585 l
S
NP 3461 3750 m 3542 3684 l
S
NP 3301 3719 m 3381 3816 l
3461 3750 l
S
NP 3221 3785 m 3301 3880 l
3381 3816 l
S
NP 3141 3850 m 3221 3944 l
3301 3880 l
S
NP 3061 3914 m 3141 4008 l
3221 3944 l
S
NP 2981 3978 m 3061 4070 l
3141 4008 l
S
NP 2900 4040 m 2980 4131 l
3061 4070 l
S
NP 2820 4101 m 2900 4190 l
2980 4131 l
S
NP 2740 4161 m 2820 4247 l
2900 4190 l
S
NP 2660 4217 m 2740 4302 l
2820 4247 l
S
NP 2580 4272 m 2660 4354 l
2740 4302 l
S
NP 2499 4322 m 2580 4402 l
2660 4354 l
S
NP 2419 4368 m 2499 4445 l
2580 4402 l
S
NP 2339 4409 m 2419 4482 l
2499 4445 l
S
NP 2259 4442 m 2339 4512 l
2419 4482 l
S
NP 2179 4466 m 2259 4532 l
2339 4512 l
S
NP 2099 4477 m 2179 4539 l
2259 4532 l
S
NP 2018 4473 m 2098 4529 l
2179 4539 l
S
NP 1938 4447 m 2018 4498 l
2098 4529 l
S
NP 1858 4392 m 1938 4436 l
2018 4498 l
S
NP 1778 4298 m 1858 4335 l
1938 4436 l
S
NP 1698 4149 m 1778 4177 l
1858 4335 l
S
NP 1618 3922 m 1698 3941 l
1778 4177 l
S
NP 1537 3585 m 1617 3593 l
1698 3941 l
S
NP 3461 3750 m 3541 3843 l
S
NP 3622 3778 m 3542 3684 l
S
NP 3541 3843 m 3622 3778 l
S
NP 3381 3816 m 3461 3907 l
3541 3843 l
S
NP 3301 3880 m 3381 3971 l
3461 3907 l
S
NP 3221 3944 m 3301 4034 l
3381 3971 l
S
NP 3141 4008 m 3221 4096 l
3301 4034 l
S
NP 3061 4070 m 3141 4157 l
3221 4096 l
S
NP 2980 4131 m 3060 4216 l
3141 4157 l
S
NP 2900 4190 m 2980 4274 l
3060 4216 l
S
NP 2820 4247 m 2900 4329 l
2980 4274 l
S
NP 2740 4302 m 2820 4382 l
2900 4329 l
S
NP 2660 4354 m 2740 4432 l
2820 4382 l
S
NP 2580 4402 m 2660 4478 l
2740 4432 l
S
NP 2499 4445 m 2579 4518 l
2660 4478 l
S
NP 2419 4482 m 2499 4553 l
2579 4518 l
S
NP 2339 4512 m 2419 4579 l
2499 4553 l
S
NP 2259 4532 m 2339 4595 l
2419 4579 l
S
NP 2179 4539 m 2259 4599 l
2339 4595 l
S
NP 2098 4529 m 2178 4585 l
2259 4599 l
S
NP 2018 4498 m 2098 4548 l
2178 4585 l
S
NP 1938 4436 m 2018 4481 l
2098 4548 l
S
NP 1858 4335 m 1938 4373 l
2018 4481 l
S
NP 1778 4177 m 1858 4209 l
1938 4373 l
S
NP 1698 3941 m 1778 3964 l
1858 4209 l
S
NP 1617 3593 m 1697 3606 l
1778 3964 l
S
NP 3541 3843 m 3622 3932 l
S
NP 3702 3868 m 3622 3778 l
S
NP 3622 3932 m 3702 3868 l
S
NP 3461 3907 m 3541 3995 l
3622 3932 l
S
NP 3381 3971 m 3461 4058 l
3541 3995 l
S
NP 3301 4034 m 3381 4120 l
3461 4058 l
S
NP 3221 4096 m 3301 4181 l
3381 4120 l
S
NP 3141 4157 m 3221 4240 l
3301 4181 l
S
NP 3060 4216 m 3140 4298 l
3221 4240 l
S
NP 2980 4274 m 3060 4354 l
3140 4298 l
S
NP 2900 4329 m 2980 4408 l
3060 4354 l
S
NP 2820 4382 m 2900 4459 l
2980 4408 l
S
NP 2740 4432 m 2820 4507 l
2900 4459 l
S
NP 2660 4478 m 2740 4551 l
2820 4507 l
S
NP 2579 4518 m 2659 4589 l
2740 4551 l
S
NP 2499 4553 m 2579 4621 l
2659 4589 l
S
NP 2419 4579 m 2499 4645 l
2579 4621 l
S
NP 2339 4595 m 2419 4658 l
2499 4645 l
S
NP 2259 4599 m 2339 4657 l
2419 4658 l
S
NP 2178 4585 m 2258 4640 l
2339 4657 l
S
NP 2098 4548 m 2178 4599 l
2258 4640 l
S
NP 2018 4481 m 2098 4527 l
2178 4599 l
S
NP 1938 4373 m 2018 4413 l
2098 4527 l
S
NP 1858 4209 m 1938 4242 l
2018 4413 l
S
NP 1778 3964 m 1858 3990 l
1938 4242 l
S
NP 1697 3606 m 1777 3623 l
1858 3990 l
S
NP 3622 3932 m 3702 4017 l
S
NP 3782 3954 m 3702 3868 l
S
NP 3702 4017 m 3782 3954 l
S
NP 3541 3995 m 3621 4079 l
3702 4017 l
S
NP 3461 4058 m 3541 4141 l
3621 4079 l
S
NP 3381 4120 m 3461 4202 l
3541 4141 l
S
NP 3301 4181 m 3381 4262 l
3461 4202 l
S
NP 3221 4240 m 3301 4320 l
3381 4262 l
S
NP 3140 4298 m 3220 4377 l
3301 4320 l
S
NP 3060 4354 m 3140 4432 l
3220 4377 l
S
NP 2980 4408 m 3060 4484 l
3140 4432 l
S
NP 2900 4459 m 2980 4534 l
3060 4484 l
S
NP 2820 4507 m 2900 4580 l
2980 4534 l
S
NP 2740 4551 m 2820 4622 l
2900 4580 l
S
NP 2659 4589 m 2739 4658 l
2820 4622 l
S
NP 2579 4621 m 2659 4687 l
2739 4658 l
S
NP 2499 4645 m 2579 4708 l
2659 4687 l
S
NP 2419 4658 m 2499 4719 l
2579 4708 l
S
NP 2339 4657 m 2419 4715 l
2499 4719 l
S
NP 2258 4640 m 2339 4694 l
2419 4715 l
S
NP 2178 4599 m 2258 4649 l
2339 4694 l
S
NP 2098 4527 m 2178 4573 l
2258 4649 l
S
NP 2018 4413 m 2098 4454 l
2178 4573 l
S
NP 1938 4242 m 2018 4277 l
2098 4454 l
S
NP 1858 3990 m 1938 4019 l
2018 4277 l
S
NP 1777 3623 m 1857 3645 l
1938 4019 l
S
NP 3702 4017 m 3782 4099 l
S
NP 3862 4036 m 3782 3954 l
S
NP 3782 4099 m 3862 4036 l
S
NP 3621 4079 m 3701 4161 l
3782 4099 l
S
NP 3541 4141 m 3621 4222 l
3701 4161 l
S
NP 3461 4202 m 3541 4282 l
3621 4222 l
S
NP 3381 4262 m 3461 4341 l
3541 4282 l
S
NP 3301 4320 m 3381 4398 l
3461 4341 l
S
NP 3220 4377 m 3300 4454 l
3381 4398 l
S
NP 3140 4432 m 3220 4507 l
3300 4454 l
S
NP 3060 4484 m 3140 4558 l
3220 4507 l
S
NP 2980 4534 m 3060 4606 l
3140 4558 l
S
NP 2900 4580 m 2980 4651 l
3060 4606 l
S
NP 2820 4622 m 2900 4691 l
2980 4651 l
S
NP 2739 4658 m 2819 4725 l
2900 4691 l
S
NP 2659 4687 m 2739 4752 l
2819 4725 l
S
NP 2579 4708 m 2659 4771 l
2739 4752 l
S
NP 2499 4719 m 2579 4779 l
2659 4771 l
S
NP 2419 4715 m 2499 4773 l
2579 4779 l
S
NP 2339 4694 m 2419 4748 l
2499 4773 l
S
NP 2258 4649 m 2338 4700 l
2419 4748 l
S
NP 2178 4573 m 2258 4619 l
2338 4700 l
S
NP 2098 4454 m 2178 4496 l
2258 4619 l
S
NP 2018 4277 m 2098 4315 l
2178 4496 l
S
NP 1938 4019 m 2018 4051 l
2098 4315 l
S
NP 1857 3645 m 1937 3670 l
2018 4051 l
S
NP 3782 4099 m 3862 4179 l
S
NP 3942 4117 m 3862 4036 l
S
NP 3862 4179 m 3942 4117 l
S
NP 3701 4161 m 3781 4240 l
3862 4179 l
S
NP 3621 4222 m 3701 4300 l
3781 4240 l
S
NP 3541 4282 m 3621 4359 l
3701 4300 l
S
NP 3461 4341 m 3541 4417 l
3621 4359 l
S
NP 3381 4398 m 3461 4473 l
3541 4417 l
S
NP 3300 4454 m 3380 4528 l
3461 4473 l
S
NP 3220 4507 m 3300 4580 l
3380 4528 l
S
NP 3140 4558 m 3220 4630 l
3300 4580 l
S
NP 3060 4606 m 3140 4677 l
3220 4630 l
S
NP 2980 4651 m 3060 4720 l
3140 4677 l
S
NP 2900 4691 m 2980 4758 l
3060 4720 l
S
NP 2819 4725 m 2899 4791 l
2980 4758 l
S
NP 2739 4752 m 2819 4816 l
2899 4791 l
S
NP 2659 4771 m 2739 4833 l
2819 4816 l
S
NP 2579 4779 m 2659 4838 l
2739 4833 l
S
NP 2499 4773 m 2579 4830 l
2659 4838 l
S
NP 2419 4748 m 2499 4802 l
2579 4830 l
S
NP 2338 4700 m 2418 4751 l
2499 4802 l
S
NP 2258 4619 m 2338 4667 l
2418 4751 l
S
NP 2178 4496 m 2258 4540 l
2338 4667 l
S
NP 2098 4315 m 2178 4354 l
2258 4540 l
S
NP 2018 4051 m 2098 4085 l
2178 4354 l
S
NP 1937 3670 m 2017 3699 l
2098 4085 l
S
NP 3862 4179 m 3942 4256 l
S
NP 4022 4195 m 3942 4117 l
S
NP 3942 4256 m 4022 4195 l
S
NP 3781 4240 m 3861 4317 l
3942 4256 l
S
NP 3701 4300 m 3781 4376 l
3861 4317 l
S
NP 3621 4359 m 3701 4435 l
3781 4376 l
S
NP 3541 4417 m 3621 4492 l
3701 4435 l
S
NP 3461 4473 m 3541 4547 l
3621 4492 l
S
NP 3380 4528 m 3461 4601 l
3541 4547 l
S
NP 3300 4580 m 3380 4652 l
3461 4601 l
S
NP 3220 4630 m 3300 4701 l
3380 4652 l
S
NP 3140 4677 m 3220 4746 l
3300 4701 l
S
NP 3060 4720 m 3140 4788 l
3220 4746 l
S
NP 2980 4758 m 3060 4825 l
3140 4788 l
S
NP 2899 4791 m 2979 4856 l
3060 4825 l
S
NP 2819 4816 m 2899 4879 l
2979 4856 l
S
NP 2739 4833 m 2819 4894 l
2899 4879 l
S
NP 2659 4838 m 2739 4898 l
2819 4894 l
S
NP 2579 4830 m 2659 4886 l
2739 4898 l
S
NP 2499 4802 m 2579 4856 l
2659 4886 l
S
NP 2418 4751 m 2498 4802 l
2579 4856 l
S
NP 2338 4667 m 2418 4715 l
2498 4802 l
S
NP 2258 4540 m 2338 4585 l
2418 4715 l
S
NP 2178 4354 m 2258 4394 l
2338 4585 l
S
NP 2098 4085 m 2178 4121 l
2258 4394 l
S
NP 2017 3699 m 2097 3730 l
2178 4121 l
S
NP 3942 4256 m 4022 4332 l
S
NP 4102 4272 m 4022 4195 l
S
NP 4022 4332 m 4102 4272 l
S
NP 3861 4317 m 3941 4392 l
4022 4332 l
S
NP 3781 4376 m 3861 4451 l
3941 4392 l
S
NP 3701 4435 m 3781 4508 l
3861 4451 l
S
NP 3621 4492 m 3701 4565 l
3781 4508 l
S
NP 3541 4547 m 3621 4619 l
3701 4565 l
S
NP 3461 4601 m 3541 4672 l
3621 4619 l
S
NP 3380 4652 m 3460 4722 l
3541 4672 l
S
NP 3300 4701 m 3380 4770 l
3460 4722 l
S
NP 3220 4746 m 3300 4814 l
3380 4770 l
S
NP 3140 4788 m 3220 4855 l
3300 4814 l
S
NP 3060 4825 m 3140 4890 l
3220 4855 l
S
NP 2979 4856 m 3059 4920 l
3140 4890 l
S
NP 2899 4879 m 2979 4942 l
3059 4920 l
S
NP 2819 4894 m 2899 4955 l
2979 4942 l
S
NP 2739 4898 m 2819 4956 l
2899 4955 l
S
NP 2659 4886 m 2739 4943 l
2819 4956 l
S
NP 2579 4856 m 2659 4911 l
2739 4943 l
S
NP 2498 4802 m 2578 4854 l
2659 4911 l
S
NP 2418 4715 m 2498 4764 l
2578 4854 l
S
NP 2338 4585 m 2418 4630 l
2498 4764 l
S
NP 2258 4394 m 2338 4437 l
2418 4630 l
S
NP 2178 4121 m 2258 4160 l
2338 4437 l
S
NP 2097 3730 m 2178 3764 l
2258 4160 l
S
NP 4022 4332 m 4102 4407 l
S
NP 4182 4347 m 4102 4272 l
S
NP 4102 4407 m 4182 4347 l
S
NP 3941 4392 m 4021 4466 l
4102 4407 l
S
NP 3861 4451 m 3941 4524 l
4021 4466 l
S
NP 3781 4508 m 3861 4581 l
3941 4524 l
S
NP 3701 4565 m 3781 4636 l
3861 4581 l
S
NP 3621 4619 m 3701 4690 l
3781 4636 l
S
NP 3541 4672 m 3621 4742 l
3701 4690 l
S
NP 3460 4722 m 3540 4791 l
3621 4742 l
S
NP 3380 4770 m 3460 4838 l
3540 4791 l
S
NP 3300 4814 m 3380 4881 l
3460 4838 l
S
NP 3220 4855 m 3300 4921 l
3380 4881 l
S
NP 3140 4890 m 3220 4955 l
3300 4921 l
S
NP 3059 4920 m 3139 4983 l
3220 4955 l
S
NP 2979 4942 m 3059 5004 l
3139 4983 l
S
NP 2899 4955 m 2979 5015 l
3059 5004 l
S
NP 2819 4956 m 2899 5015 l
2979 5015 l
S
NP 2739 4943 m 2819 5000 l
2899 5015 l
S
NP 2659 4911 m 2739 4965 l
2819 5000 l
S
NP 2578 4854 m 2658 4906 l
2739 4965 l
S
NP 2498 4764 m 2578 4814 l
2658 4906 l
S
NP 2418 4630 m 2498 4677 l
2578 4814 l
S
NP 2338 4437 m 2418 4480 l
2498 4677 l
S
NP 2258 4160 m 2338 4200 l
2418 4480 l
S
NP 2178 3764 m 2258 3800 l
2338 4200 l
S
NP 4102 4407 m 4182 4480 l
S
NP 4262 4420 m 4182 4347 l
S
NP 4182 4480 m 4262 4420 l
S
NP 4021 4466 m 4101 4538 l
4182 4480 l
S
NP 3941 4524 m 4021 4596 l
4101 4538 l
S
NP 3861 4581 m 3941 4652 l
4021 4596 l
S
NP 3781 4636 m 3861 4707 l
3941 4652 l
S
NP 3701 4690 m 3781 4760 l
3861 4707 l
S
NP 3621 4742 m 3701 4811 l
3781 4760 l
S
NP 3540 4791 m 3620 4860 l
3701 4811 l
S
NP 3460 4838 m 3540 4905 l
3620 4860 l
S
NP 3380 4881 m 3460 4948 l
3540 4905 l
S
NP 3300 4921 m 3380 4986 l
3460 4948 l
S
NP 3220 4955 m 3300 5019 l
3380 4986 l
S
NP 3139 4983 m 3219 5046 l
3300 5019 l
S
NP 3059 5004 m 3139 5065 l
3219 5046 l
S
NP 2979 5015 m 3059 5075 l
3139 5065 l
S
NP 2899 5015 m 2979 5073 l
3059 5075 l
S
NP 2819 5000 m 2899 5056 l
2979 5073 l
S
NP 2739 4965 m 2819 5020 l
2899 5056 l
S
NP 2658 4906 m 2738 4959 l
2819 5020 l
S
NP 2578 4814 m 2658 4864 l
2738 4959 l
S
NP 2498 4677 m 2578 4725 l
2658 4864 l
S
NP 2418 4480 m 2498 4525 l
2578 4725 l
S
NP 2338 4200 m 2418 4242 l
2498 4525 l
S
NP 2258 3800 m 2338 3838 l
2418 4242 l
S
NP 4182 4480 m 4262 4552 l
S
NP 4342 4493 m 4262 4420 l
S
NP 4262 4552 m 4342 4493 l
S
NP 4101 4538 m 4181 4610 l
4262 4552 l
S
NP 4021 4596 m 4101 4667 l
4181 4610 l
S
NP 3941 4652 m 4021 4722 l
4101 4667 l
S
NP 3861 4707 m 3941 4777 l
4021 4722 l
S
NP 3781 4760 m 3861 4829 l
3941 4777 l
S
NP 3701 4811 m 3781 4879 l
3861 4829 l
S
NP 3620 4860 m 3700 4927 l
3781 4879 l
S
NP 3540 4905 m 3620 4972 l
3700 4927 l
S
NP 3460 4948 m 3540 5013 l
3620 4972 l
S
NP 3380 4986 m 3460 5051 l
3540 5013 l
S
NP 3300 5019 m 3380 5083 l
3460 5051 l
S
NP 3219 5046 m 3300 5108 l
3380 5083 l
S
NP 3139 5065 m 3219 5127 l
3300 5108 l
S
NP 3059 5075 m 3139 5135 l
3219 5127 l
S
NP 2979 5073 m 3059 5132 l
3139 5135 l
S
NP 2899 5056 m 2979 5113 l
3059 5132 l
S
NP 2819 5020 m 2899 5075 l
2979 5113 l
S
NP 2738 4959 m 2818 5012 l
2899 5075 l
S
NP 2658 4864 m 2738 4915 l
2818 5012 l
S
NP 2578 4725 m 2658 4774 l
2738 4915 l
S
NP 2498 4525 m 2578 4572 l
2658 4774 l
S
NP 2418 4242 m 2498 4285 l
2578 4572 l
S
NP 2338 3838 m 2418 3879 l
2498 4285 l
S
NP 4262 4552 m 4342 4623 l
S
NP 4422 4564 m 4342 4493 l
S
NP 4342 4623 m 4422 4564 l
S
NP 4181 4610 m 4261 4680 l
4342 4623 l
S
NP 4101 4667 m 4181 4737 l
4261 4680 l
S
NP 4021 4722 m 4101 4792 l
4181 4737 l
S
NP 3941 4777 m 4021 4845 l
4101 4792 l
S
NP 3861 4829 m 3941 4897 l
4021 4845 l
S
NP 3781 4879 m 3861 4947 l
3941 4897 l
S
NP 3700 4927 m 3780 4994 l
3861 4947 l
S
NP 3620 4972 m 3700 5038 l
3780 4994 l
S
NP 3540 5013 m 3620 5079 l
3700 5038 l
S
NP 3460 5051 m 3540 5115 l
3620 5079 l
S
NP 3380 5083 m 3460 5146 l
3540 5115 l
S
NP 3300 5108 m 3380 5171 l
3460 5146 l
S
NP 3219 5127 m 3299 5188 l
3380 5171 l
S
NP 3139 5135 m 3219 5195 l
3299 5188 l
S
NP 3059 5132 m 3139 5190 l
3219 5195 l
S
NP 2979 5113 m 3059 5170 l
3139 5190 l
S
NP 2899 5075 m 2979 5131 l
3059 5170 l
S
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14 /Times-Roman-Maple fsf
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%%306 72 (Page 1) pagenum
showpage
%%Trailer
%% END of MapleV Session Output

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